Constructions and uses of incomplete pairwise balanced designs

We give explicit constructions for incomplete pairwise balanced designs IPBD((v; w), K), or, equivalently, edge-decompositions of a difference of two cliques Kv\Kw\ into cliques whose sizes belong to the set K. Our constructions produce such designs whenever v and w satisfy the usual divisibility conditions, have ratio v / w bounded away from the smallest value in K minus one, say v/w>k-1+epsilon, for k=min K and epsilon>0, and are sufficiently large (depending on K and epsilon. As a consequence, some new results are obtained on many related designs, including class-uniformly resolvable designs, incomplete mutually orthogonal latin squares, and group divisible designs. We also include several other applications that illustrate the power of using IPBDs as 'templates'.